DIFFRACTION by ewghwehws


									                 CHAPTER 3
                IN CRYSTAL
                   I.     X-Ray
                   II.    Diffraction
                   III.   Diffraction of Waves by
                   IV.    X-Ray Diffraction
                   V.     Bragg Equation
                   VI.    X-Ray Methods
Bertha Röntgen’s   VII.   Neutron & Electron Diffraction
Hand 8 Nov, 1895

   X-rays were discovered in
    1895     by  the     German
    physicist Wilhelm Conrad
    Röntgen and were so named
    because their nature was
    unknown at the time.

    He was awarded the Nobel
    prize for physics in 1901.
                                  Wilhelm Conrad Röntgen


   X ray, invisible, highly penetrating electromagnetic radiation of
    much shorter wavelength (higher frequency) than visible light.
    The wavelength range for X rays is from about 10-8 m to about
    10-11 m, the corresponding frequency range is from about 3 ×
    1016 Hz to about 3 × 1019 Hz.

          X-RAY ENERGY
   Electromagnetic radiation described as having packets of
    energy, or photons. The energy of the photon is related to
    its frequency by the following formula:

                                                         E  h
                                          hc                     c
                                                           

 =Wavelength , ‫ = ע‬Frequency , c = Velocity of light     E 

        x-ray ≈   10-10 ≈ 1A°                     E ~ 104 ev


   Visible light photons and X-ray photons are both
    produced by the movement of electrons in atoms.
    Electrons occupy different energy levels, or orbitals,
    around an atom's nucleus.

    When an electron drops to a lower orbital, it needs to
    release some energy; it releases the extra energy in the
    form of a photon. The energy level of the photon depends
    on how far the electron dropped between orbitals.

          X-RAY TUBE
   X rays can be produced in a highly evacuated glass bulb, called
    an X-ray tube, that contains essentially two electrodes—an
    anode made of platinum, tungsten, or another heavy metal of
    high melting point, and a cathode. When a high voltage is
    applied between the electrodes, streams of electrons (cathode
    rays) are accelerated from the cathode to the anode and
    produce X rays as they strike the anode.
                   Evacuated glass bulb


         Monochromatic and Broad
         Spectrum of X-rays
   X-rays can be created by bombarding a metal target with
    high energy (> 104 ) electrons.

   Some of these electrons excite electrons from core states in
    the metal, which then recombine, producing highly
    monochromatic X-rays. These are referred to as
    characteristic X-ray lines.

   Other electrons, which are decelerated by the periodic
    potential of the metal, produce a broad spectrum of X-ray

   Depending on the diffraction experiment, either or both of
    these X-ray spectra can be used.      10 4


   The atoms that make up your body         ...something
    tissue absorb visible light photons        you won't
    very well. The energy level of the           see very
    photon fits with various energy               often
    differences   between       electron         (Visible
    positions.                                    Light)

   Radio waves don't have enough
    energy to move electrons between
    orbitals in larger atoms, so they pass
    through most stuff. X-ray photons also      X-ray
    pass through most things, but for the
    opposite reason: They have too much

         Generation of X-rays (K-Shell
    An electron in a higher orbital immediately falls to the lower
energy level, releasing its extra energy in the form of a photon. It's
a big drop, so the photon has a high energy level; it is an X-ray

                                            The free electron collides
                                             with the tungsten atom,
                                          knocking an electron out of a
                                          lower orbital. A higher orbital
                                             electron fills the empty
                                          position, releasing its excess
                                              energy as a photon.

      Absorption of X-rays

   A larger atom is more likely to absorb an X-ray photon in
    this way, because larger atoms have greater energy
    differences between orbitals -- the energy level more
    closely matches the energy of the photon. Smaller atoms,
    where the electron orbitals are separated by relatively low
    jumps in energy, are less likely to absorb X-ray photons.

   The soft tissue in your body is composed of smaller
    atoms, and so does not absorb X-ray photons particularly
    well. The calcium atoms that make up your bones are
    much larger, so they are better at absorbing X-ray


   Diffraction is a wave phenomenon in
    which the apparent bending and
    spreading of waves when they meet an
   Diffraction occurs with electromagnetic
    waves, such as light and radio waves,
    and also in sound waves and water
                                                   Width b Variable
                                                    (500-1500 nm)
   The most conceptually simple example         Wavelength Constant
    of diffraction is double-slit diffraction,
                                                      (600 nm)
    that’s why firstly we remember light
    diffraction.                                 Distance d = Constant


   Light diffraction is caused by light bending around the edge of
    an object. The interference pattern of bright and dark lines from
    the diffraction experiment can only be explained by the additive
    nature of waves; wave peaks can add together to make a
    brighter light, or a peak and a through will cancel each other out
    and result in darkness.

    Thus Young’s light interference
     experiment proves that light
       has wavelike properties.


          Constructive & Destructive Waves

   Constructive interference is        Destructive İnterference     .
    the result of synchronized           results when two out-of-phase
    light   waves     that    add        light waves cancel each other
    together to increase the light       out, resulting in darkness.

Light Interference

         Diffraction from a particle and solid

Single particle
 To understand diffraction we also
  have to consider what happens when
  a wave interacts with a single particle.
  The particle scatters the incident
  beam uniformly in all directions
Solid material
 What happens if the beam is
  incident on solid material? If we
  consider a crystalline material, the
  scattered beams may add together
  in a few directions and reinforce
  each other to give diffracted beams
          Diffraction of Waves by Crystals

                 A crystal is a periodic structure
              ( unit cells are repeated regularly)

    Solid State Physics deals how the waves are propagated
 through such periodic structures. In this chapter we study the
 crystal structure through the diffraction of photons (X-ray),
 nuetrons and electrons.

                   X-ray     Neutron      Electron

The general princibles will be the same for each type of waves.
          Diffraction of Waves by Crystals

   The diffraction depends on the crystal structure and on
    the wavelength.
   At optical wavelengths such as 5000 angstroms the
    superposition of the waves scattered elastically by the
    individual atoms of a crystal results in ordinary optical
   When the wavelength of the radiation is comparable
    with or smaller than the lattice constant, one can find
    diffracted beams in directions quite different from the
    incident radiation.

         Diffraction of Waves by Crystals

   The structure of a crystal can be determined by
    studying the diffraction pattern of a beam of radiation
    incident on the crystal.

   Beam diffraction takes place only in certain specific
    directions, much as light is diffracted by a grating.

   By measuring the directions of the diffraction and the
    corresponding intensities, one obtains information
    concerning the crystal structure responsible for

   X-ray crystallography is a technique in crystallography in
    which the pattern produced by the diffraction of x-rays through
    the closely spaced lattice of atoms in a crystal is recorded and
    then analyzed to reveal the nature of that lattice.

   X-ray diffraction = (XRD)

         X-Ray Crystallography

   The wavelength of X-rays is
    typically 1 A°, comparable to the
    interatomic spacing (distances
    between atoms or ions) in solids.

   We need X-rays:

                         hc   hc
     Exray      h           12.3x103 eV
                          1x10 m

          Crystal Structure Determination

       A crystal behaves as a 3-D diffraction grating for x-rays
   In a diffraction experiment, the spacing of lines on the grating
    can be deduced from the separation of the diffraction maxima
    Information about the structure of the lines on the
    grating can be obtained by measuring the relative
    intensities of different orders
   Similarly, measurement of the separation of the X-ray
    diffraction maxima from a crystal allows us to determine
    the size of the unit cell and from the intensities of
    diffracted beams one can obtain information about the
    arrangement of atoms within the cell.
    X-Ray Diffraction

    W. L. Bragg presented a simple
explanation of the diffracted beams from a
    The Bragg derivation is simple but is
convincing only since it reproduces the
correct result.

          X-Ray Diffraction & Bragg Equation

   English physicists Sir W.H. Bragg
    and his son Sir W.L. Bragg
    developed a relationship in 1913 to
    explain why the cleavage faces of
    crystals appear to reflect X-ray
    beams at certain angles of incidence
    (theta, θ).This observation is an          Sir William Henry Bragg (1862-1942),
    example of X-ray wave interference.        William Lawrence Bragg (1890-1971)

o 1915, the father and son were awarded the Nobel prize for physics
    "for their services in the analysis of crystal structure by means of

        Bragg Equation
   Bragg law identifies the angles of the incident
    radiation relative to the lattice planes for which
    diffraction peaks occurs.
   Bragg derived the condition for constructive
    interference of the X-rays scattered from a set of
    parallel lattice planes.

   W.L. Bragg considered crystals to be made up of parallel
    planes of atoms. Incident waves are reflected specularly from
    parallel planes of atoms in the crystal, with each plane is
    reflecting only a very small fraction of the radiation, like a
    lightly silvered mirror.
   In mirrorlike reflection the angle of incidence is equal to the
    angle of reflection.

                          ө              ө

       Diffraction Condition

 The diffracted beams are found to occur
  when the reflections from planes of atoms
  interfere constructively.
 We treat     elastic scattering, in which the
  energy of X-ray is not changed on reflection.

          Bragg Equation
   When the X-rays strike a layer of a crystal, some of them will
    be reflected. We are interested in X-rays that are in-phase
    with one another. X-rays that add together constructively in x-
    ray diffraction analysis in-phase before they are reflected and
    after they reflected.

                                               Incident angle
                                               Reflected angle
                                                Wavelength of X-ray
                                 2

                                                Total Diffracted
                                                    Angle  2
                 Bragg Equation

   These two x-ray beams travel slightly different distances. The
    difference in the distances traveled is related to the distance
    between the adjacent layers.
   Connecting the two beams with perpendicular lines shows the
    difference between the top and the bottom beams.

                                                The line CE is equivalent
                                                 to the distance between
                                                     the two layers (d)

                                                      DE  d sin

          Bragg Law
   The length DE is the same as EF, so the total distance
    traveled by the bottom wave is expressed by:

                                                 EF  d sin
                                                DE  d sin

                                                DE  EF  2d sin
                                                 n  2d sin 

   Constructive interference of the radiation from successive
    planes occurs when the path difference is an integral
    number of wavelenghts. This is the Bragg Law.
          Bragg Equation
                     2d sin  n
where, d is the spacing of the planes and n is the order of diffraction.

   Bragg reflection can only occur for wavelength

                       n  2d
   This is why we cannot use visible light. No diffraction occurs when
    the above condition is not satisfied.

    The diffracted beams (reflections) from any set of lattice planes
    can only occur at particular angles pradicted by the Bragg law.
       Scattering of X-rays from adjacent
             lattice points A and B
     X-rays are incident at an angle  on one of the planes
of the set.
     There will be constructive interference of the waves
scattered from the two successive lattice points A and B in
the plane if the distances AC and DB are equal.

                       D         C

                                       2
                   A                 B
          Constructive interference of waves
            scattered from the same plane
If the scattered wave makes the same angle to the plane as
                       the incident wave

The diffracted wave looks as if it has been reflected from the

     We consider the scattering from lattice points rather
 than atoms because it is the basis of atoms associated with
 each lattice point that is the true repeat unit of the crystal;
 The lattice point is analoque of the line on optical diffraction
 grating and the basis represents the structure of the line.

    Diffraction maximum
     Coherent scattering from a single plane is not
sufficient to obtain a diffraction maximum. It is also
necessary that successive planes should scatter
in phase

 This will be the case if the path difference for
 scattering off two adjacent planes is an integral
              number of wavelengths

               2d sin  n
         Labelling the reflection planes

   To label the reflections, Miller indices of the planes
    can be used.
   A beam corresponding to a value of n>1 could be
    identified by a statement such as ‘the nth-order
    reflections from the (hkl) planes’.
   (nh nk nl) reflection
             Third-order reflection from (111) plane

                        (333) reflection
              n-th order diffraction off (hkl)
   Rewriting the Bragg law
                    d 
                   2  sin   
    which makes n-th order diffraction off (hkl) planes of
    spacing ‘d’ look like first-order diffraction off planes
    of spacing d/n.

   Planes of this reduced spacing would have Miller
    indices (nh nk nl).

           X-ray structure analysis of NaCl
                       and KCl
            DEDUCE the STRUCTURE of NaCl and KCl
 Bragg used an ordinary spectrometer and measured the intensity of
          specular reflection from a cleaved face of a crystal

    found six values of  for which a sharp peak in intensity occurred,
      corresponding to three characteristics wavelengths (K,L and M x-
          rays) in first and second order (n=1 and n=2 in Bragg law)
   By repeating the experiment with a different crystal face he could
    use his eqn. to find for example the ratio of (100) and (111) plane
    spacings, information that confirmed the cubic symmetry of the
    atomic arrangement.
           Details of structure
      Details of structure were than deduced from the differences
    between the diffraction patterns for NaCl and KCl.
   Major difference; absence of (111) reflection in KCl compared to a
    weak but detectable (111) reflection in NaCl.

   This arises because the K and Cl ions both
    have the argon electron shell structure and
    hence scatter x-rays almost equally whereas
    Na and Cl ions have different scattering
    strengths. (111) reflection in NaCl corresponds
    to one wavelength of path difference between
    neighbouring (111) planes.

             Experimental arrangements
                for x-ray diffraction

   Since the pioneering work of Bragg, x-ray
    diffraction has become into a routine
    technique for the determination of crsytal

    Bragg Equation
   Since Bragg's Law applies to all sets of crystal planes,
the lattice can be deduced from the diffraction pattern,
making use of general expressions for the spacing of the
planes in terms of their Miller indices. For cubic structures

                           h2  k 2  l 2

    Note that the smaller the spacing the higher the angle
of diffraction, i.e. the spacing of peaks in the diffraction
pattern is inversely proportional to the spacing of the planes
in the lattice. The diffraction pattern will reflect the
symmetry properties of the lattice.
                    2d sin   n
Bragg Equation
           A simple example is the difference between
     the series of (n00) reflections for a simple
     cubic and a body centred cubic lattice. For the
     simple cubic lattice, all values of n will give Bragg
            However, for the body centred cubic lattice
     the (100) planes are interleaved by an equivalent
     set at the halfway position. At the angle where
     Bragg's Law would give the (100) reflection the
     interleaved planes will give a reflection exactly out
     of phase with that from the primary planes, which
     will exactly cancel the signal. There is no signal
     from (n00) planes with odd values of n. This kind
     of argument leads to rules for identifying the
     lattice symmetry from "missing" reflections, which
     are often quite simple.
         Types of X-ray camera

         There are many types of X-ray camera to
     sort out reflections from different crystal
     planes. We will study only three types of X-ray
     photograph that are widely used for the simple
1.   Laue photograph
2.   Rotating crystal method
3.   Powder photograph


                X-Ray Diffraction Method

      Laue           Rotating Crystal           Powder

    Orientation        Lattice constant     Lattice Parameters
   Single Crystal       Single Crystal    Polycrystal (powdered)
Polychromatic Beam   Monochromatic Beam   Monochromatic Beam
    Fixed Angle         Variable Angle        Variable Angle

               LAUE METHOD
       The Laue method is mainly used to determine the
        orientation of large single crystals while radiation is
        reflected from, or transmitted through a fixed crystal.

       The diffracted beams form arrays of
        spots, that lie on curves on the film.

        The Bragg angle is fixed for every
        set of planes in the crystal. Each set
        of planes picks out and diffracts the
        particular wavelength from the white
        radiation that satisfies the Bragg law
        for the values of d and θ involved.
         Back-reflection Laue Method

   In the back-reflection method, the film is placed between the
    x-ray source and the crystal. The beams which are diffracted
    in a backward direction are recorded.

   One side of the cone of Laue
    reflections is defined by the
    transmitted beam. The film
    intersects the cone, with the
    diffraction spots generally lying
    on an hyperbola.                                 Single
                                   X-Ray     Film

        Transmission Laue Method

   In the transmission Laue method, the film is placed behind
    the crystal to record beams which are transmitted through
    the crystal.

   One side of the cone of Laue
    reflections is defined by the
    transmitted beam. The film
    intersects the cone, with the
    diffraction spots generally
    lying on an ellipse.
                                    Single         Film
Laue Pattern        The symmetry of the
                spot pattern reflects the
                symmetry of the crystal
                when viewed along the
                direction of the incident
                beam. Laue method is
                often used to determine
                the orientation of single
                crystals by means of
                illuminating the crystal
                with a continuos spectrum
                of X-rays;
                Single crystal
                Continous spectrum of x-
48              Symmetry of the crystal;
       Crystal structure
       determination by Laue method
   Therefore, the Laue method is mainly used to
    determine the crystal orientation.
   Although the Laue method can also be used to
    determine the crystal structure, several
    wavelengths can reflect in different orders from
    the same set of planes, with the different order
    reflections superimposed on the same spot in
    the film. This makes crystal structure
    determination by spot intensity diffucult.
   Rotating crystal method overcomes this
    problem. How?

   In the rotating crystal method, a
    single crystal is mounted with
    an     axis    normal     to    a
    monochromatic        x-ray beam.
    A cylindrical film is placed
    around it and the crystal is
    rotated about the chosen axis.

       As the crystal rotates, sets of lattice planes will at some
        point make the correct Bragg angle for the monochromatic
        incident beam, and at that point a diffracted beam will be
    Lattice constant of the crystal can be
determined by means of this method; for a
given wavelength if the angle  at which a
reflection occurs is known,    d hkl can be
          d 
                h2  k 2  l 2

      Rotating Crystal Method

    The reflected beams are located on the surface of
imaginary cones. By recording the diffraction patterns (both
angles and intensities) for various crystal orientations, one
can determine the shape and size of unit cell as well as
arrangement of atoms inside the cell.



     If a powdered specimen is used, instead of a
single crystal, then there is no need to rotate
the specimen, because there will always be
some crystals at an orientation for which
diffraction is permitted. Here a monochromatic
X-ray beam is incident on a powdered or
polycrystalline sample.
     This method is useful for samples that are
difficult to obtain in single crystal form.


    The powder method is used to determine the value
of the lattice parameters accurately. Lattice parameters
are the magnitudes of the unit vectors a, b and c which
define the unit cell for the crystal.

     For every set of crystal planes, by chance, one or
more crystals will be in the correct orientation to give
the correct Bragg angle to satisfy Bragg's equation.
Every crystal plane is thus capable of diffraction. Each
diffraction line is made up of a large number of small
spots, each from a separate crystal. Each spot is so
small as to give the appearance of a continuous line.
           The Powder Method

      sample of consists of some
    A the samplesome x-ray beam
    If a monochromatic hundreds of
    tens of (i.e. a a powdered
                  at     single crystal,
    is directedrandomly orientated
    single crystals, two
    sample) show or the
    then only one thatthe diffracted
             form seen
             are continuous cones.
    beams may result. to lie on the
    surface of film is used to record
    A circle of several cones. The
    the diffraction emerge in all
    cones may pattern as shown.
    Each cone        forwards
    directions, intersects the and  film
    giving diffraction lines. The lines
    are seen as arcs on the film.

     Debye Scherrer Camera
   A very small amount of powdered material is sealed
into a fine capillary tube made from glass that does not
diffract x-rays.

   The specimen is placed
in the Debye Scherrer
camera and is accurately
aligned to be in the centre
of the camera. X-rays enter
the camera through a

      Debye Scherrer Camera

    The powder diffracts
the x-rays in accordance
with     Braggs     law   to
produce        cones      of
diffracted beams. These
cones intersect a strip of
photographic film located
in the cylindrical camera to
produce a characteristic
set of arcs on the film.

    Powder diffraction film

      When the film is removed from the camera,
flattened and processed, it shows the diffraction
lines and the holes for the incident and
transmitted beams.

          Application of XRD
         XRD is a nondestructive technique. Some of the uses of
     x-ray diffraction are;

1.     Differentiation between crystalline and amorphous
2.     Determination of the structure of crystalline materials;
3.     Determination of electron distribution within the atoms, and
       throughout the unit cell;
4.     Determination of the orientation of single crystals;
5.     Determination of the texture of polygrained materials;
6.     Measurement of strain and small grain size…..etc

Advantages and
disadvantages of X-rays
    X-ray is the cheapest, the most convenient and
     widely used method.
    X-rays are not absorbed very much by air, so
     the specimen need not be in an evacuated
    They do not interact very strongly with lighter

   Difraction Methods

     X-ray          Neutron          Electron

   Different radiation source of neutron or
electron can       also be used in diffraction
    The physical basis for the diffraction of
electron and neutron beams is the same as that
for the diffraction of X rays, the only difference
being in the mechanism of scattering.
      Neutron Diffraction

   Neutrons were discovered in 1932 and their wave
    properties was shown in 1936.
                  E = p2/2m        p = h/λ
                   E=Energy λ=Wavelength
                mn=Mass of neutron = 1,67.10-27kg

   λ ~1A°; Energy E~0.08 eV. This energy is of the same
    order of magnitude as the thermal energy kT at room
    temperature, 0.025 eV, and for this reason we speak of
    thermal neutrons.

      Neutron Diffraction

   Neutron does not interact with electrons in the crystal.
    Thus, unlike the x-ray, which is scattered entirely by
    electrons, the neutron is scattered entirely by nuclei

   Although uncharged, neutron has an intrinsic magnetic
    moment, so it will interact strongly with atoms and ions
    in the crystal which also have magnetic moments.

   Neutrons are more useful than X-rays for determining
    the crystal structures of solids containing light

   Neutron sources in the world are limited so neutron
    diffraction is a very special tool.
          Neutron Diffraction

        Neutron diffraction has several advantages over its x-
    ray counterpart;
   Neutron diffraction is an important tool in the investigation
    of magnetic ordering that occur in some materials.

   Light atoms such as H are better resolved in a neutron
    pattern because, having only a few electrons to scatter
    the X ray beam, they do not contribute significantly to the
    X ray diffracted pattern.

      Electron Diffraction
     Electron diffraction has also been used in the analysis of
crystal structure. The electron, like the neutron, possesses wave

      2k 2
                    40eV                  2A     0
     2me 2me    2

      Electrons are charged particles and interact strongly with
all atoms. So electrons with an energy of a few eV would be
completely absorbed by the specimen. In order that an
electron beam can penetrate into a specimen , it necessitas a
beam of very high energy (50 keV to 1MeV) as well as the
specimen must be thin (100-1000 nm)
       Electron Diffraction

     If low electron energies are used, the penetration depth
will be very small (only about 50 A°), and the beam will be
reflected from the surface. Consequently, electron diffraction is
a useful technique for surface structure studies.

    Electrons are scattered strongly in air, so diffraction
experiment must be carried out in a high vacuum. This brings
complication and it is expensive as well.

          Diffraction Methods

     X-Ray                 Neutron                Electron

       λ = 1A°                λ = 1A°                  λ = 2A°

     E ~ 104 eV              E ~ 0.08 eV             E ~ 150 eV

interact with electron   interact with nuclei   interact with electron
     Penetrating         Highly Penetrating       Less Penetrating


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